marginals: Simple and double intergrals for calculation of main effects

In mcrucifix/gp: Gaussian process calibration for emulation

Simple and double intergrals for calculation of main effects

Description

Estimate simple and double integrals for estimators of marginal mens and variances calculations for Gaussian process main effect and associated variance analysis

Usage

marginals(E, rhoarray, p, method='deterministic')

Arguments

E	A Gaussian process list as generated by GP_C

rhoarray	An array of n1 x n2 x n3 ... with as many dimensions as the number of design factors used to generate the Gaussian process. The array represent the discretized density function. The densitity must be associated with a list of levels correspnding to the mid-points of the distribution. See examples.
p	A vector of the indices over which the integral of squares are being estimated (see details)
method	The determistic method assuming a sum over all components of rhoarray is currently the only one supported.

Details

varanal will estimate the emulator variances (more precisely, integral of squared quantities !) associated with the variance of indices complementary to p (-p in the notation of Oakley and Ohagan, 2004, p. 761), and then take the expectation of this quantity over p. the less indices in "p", the more computationally intensive given that all (-p) x (-p) combinations have to be summed up. variances measures will then be obtained as follows,

total sensitivty index associated with, e.g., vars 1 and 2 (assuming 5 factors in total: varanal(E, rho, c(1,2,3,4,5)) - varanal(E, rho, c(3,4,5))

mean sensitivty index associated with the same variables varanal(E, rho, c(1,2) ) - varanal (E, rho, NULL )

Incidentially, varanal(E, rho, c(1,2,3,4,5)) - varanal(E, rho, NULL) will return the total variance.

Value

A list with following elements:

simple_integrals

Matrix with as many columns as levels supplied along p with lines corresponding to simple integrals Simple integrals over -p (the complementary indices to p), of y: the GP mean, and its regression and stochastic components, y_m and y_g, respectively.

simple_integrals

Matrix with as many columns as levels supplied along p with lines corresponding to double integrals over -p x -p, composed of vectors with as many elements as levels supplied along p with elements corresponding to double integrals over Sp: the GP co-variance, yyt: the GP mean products, and its contributions associated with the regression, co-variance between regression and stochastic components, and stocastic compenents respectively: yyt_m, yyt_mg, yyt_g.

Author(s)

Michel Crucifix

See Also

varanal

Examples

# generate data
X = as.matrix( expand.grid(seq(3), seq(3), seq(3))) 
Y = apply (X, 1, function(x) { x[1] - 2*x[1]*x[2] + x[3] - 2} )
Y = Y + rnorm(length(Y), sd=0.1)

E = GP_C(X, Y, lambda=list(theta=c(1,1,1), nugget=0.1))

# define rho function : uniform distribution

n = 8

levels = list ( seq(0,3, l=n), seq(0,3, l=n), seq(0,3, l=n) )
rho    = array(1, c(n,n,n))
rho    = rho / sum(rho) # not necessary. 
attr(rho, "levels") <- levels  # this is necessary ! 

# mean sensitivity index associated with the index 2

E_2 = marginals(E, rho,  c(2))

plot(E_2$simple_integrals['y',], type='l', main='Main effect of factor 2')
lines(E_2$simple_integrals['y',] +sqrt ( E_2$double_integrals['Sp',]  ), 
   lty=2, col='blue')

# variances associated with Gaussian process variance

lines(E_2$simple_integrals['y',] -sqrt ( E_2$double_integrals['Sp',]  ), 
   lty=2, col='blue')


# variances associated with process mean : 

lines(E_2$simple_integrals['y',] +sqrt ( 
    E_2$simple_integrals['yyt',]  - E_2$simple_integrals['y',]^2 ), 
   lty=2, col='red')

lines(E_2$simple_integrals['y',] -sqrt ( 
    E_2$simple_integrals['yyt',]  - E_2$simple_integrals['y',]^2 ), 
   lty=2, col='red')



legend('bottomleft', c('sd associated with GP variance', 'sd associated with process mean'), 
                 lty=2, col=c('red','blue'))

mcrucifix/gp documentation built on July 29, 2023, 8:58 p.m.